Plus, the postcard is "from the Donut Islands"--why would it be in Tim's apartment at all if "she hasn't sent it"? If Isabella wanted to send MC a postcard from the Donut Islands, she would
send it from the Donut Islands. It makes absolutely no sense that she would buy one there
and carry it home to mail it, and less sense yet that having done so, she would stick it in a vase and forget about it. The entire thing is utterly insupportable.
I think you're making a big deal over something so minor dude (you're not the only one mind you).
Maybe Izzy filled it out, was ready to send it and simply forgot to (for whatever reason) from the Donut Islands.
I've bought heaps of postcards from holiday destinations, but kept them because I was in a specific town
and I liked to have them as a memory of places I've been to.
So maybe Izzy decided not to send it due to her and the MC putting their relationship on hold, or ending it altogether, and decided against sending it to him, as to not complicate things further.
Tim, upon tidying up the bills, paperwork and said postcard, threw all of that into the vase,
didn't notice the postcard, all of this whilst Izzy was away overseas for work.
Whenever, such a "plot-hole" comes up, I always think of my own way to figure out an answer
for why such a scene is possible. And usually it's not that difficult.